MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isobs Structured version   Unicode version

Theorem isobs 18143
Description: The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
isobs.v  |-  V  =  ( Base `  W
)
isobs.h  |-  .,  =  ( .i `  W )
isobs.f  |-  F  =  (Scalar `  W )
isobs.u  |-  .1.  =  ( 1r `  F )
isobs.z  |-  .0.  =  ( 0g `  F )
isobs.o  |-  ._|_  =  ( ocv `  W )
isobs.y  |-  Y  =  ( 0g `  W
)
Assertion
Ref Expression
isobs  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) )
Distinct variable groups:    x, y,  .,    x,  .0. , y    x,  .1. , y    x, B, y   
x, W, y
Allowed substitution hints:    F( x, y)    ._|_ ( x, y)    V( x, y)    Y( x, y)

Proof of Theorem isobs
Dummy variables  h  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-obs 18128 . . . . 5  |- OBasis  =  ( h  e.  PreHil  |->  { b  e.  ~P ( Base `  h )  |  ( A. x  e.  b 
A. y  e.  b  ( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  /\  ( ( ocv `  h ) `  b
)  =  { ( 0g `  h ) } ) } )
21dmmptss 5332 . . . 4  |-  dom OBasis  C_  PreHil
3 elfvdm 5714 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  dom OBasis )
42, 3sseldi 3352 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
5 fveq2 5689 . . . . . . . . 9  |-  ( h  =  W  ->  ( Base `  h )  =  ( Base `  W
) )
6 isobs.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
75, 6syl6eqr 2491 . . . . . . . 8  |-  ( h  =  W  ->  ( Base `  h )  =  V )
87pweqd 3863 . . . . . . 7  |-  ( h  =  W  ->  ~P ( Base `  h )  =  ~P V )
9 fveq2 5689 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( .i `  h )  =  ( .i `  W
) )
10 isobs.h . . . . . . . . . . . 12  |-  .,  =  ( .i `  W )
119, 10syl6eqr 2491 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( .i `  h )  = 
.,  )
1211oveqd 6106 . . . . . . . . . 10  |-  ( h  =  W  ->  (
x ( .i `  h ) y )  =  ( x  .,  y ) )
13 fveq2 5689 . . . . . . . . . . . . . 14  |-  ( h  =  W  ->  (Scalar `  h )  =  (Scalar `  W ) )
14 isobs.f . . . . . . . . . . . . . 14  |-  F  =  (Scalar `  W )
1513, 14syl6eqr 2491 . . . . . . . . . . . . 13  |-  ( h  =  W  ->  (Scalar `  h )  =  F )
1615fveq2d 5693 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  ( 1r `  F ) )
17 isobs.u . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  F )
1816, 17syl6eqr 2491 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  .1.  )
1915fveq2d 5693 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  ( 0g `  F ) )
20 isobs.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  F )
2119, 20syl6eqr 2491 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  .0.  )
2218, 21ifeq12d 3807 . . . . . . . . . 10  |-  ( h  =  W  ->  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  =  if ( x  =  y ,  .1.  ,  .0.  )
)
2312, 22eqeq12d 2455 . . . . . . . . 9  |-  ( h  =  W  ->  (
( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  <-> 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )
) )
24232ralbidv 2755 . . . . . . . 8  |-  ( h  =  W  ->  ( A. x  e.  b  A. y  e.  b 
( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  <->  A. x  e.  b  A. y  e.  b 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )
) )
25 fveq2 5689 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
26 isobs.o . . . . . . . . . . 11  |-  ._|_  =  ( ocv `  W )
2725, 26syl6eqr 2491 . . . . . . . . . 10  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
2827fveq1d 5691 . . . . . . . . 9  |-  ( h  =  W  ->  (
( ocv `  h
) `  b )  =  (  ._|_  `  b
) )
29 fveq2 5689 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  h )  =  ( 0g `  W
) )
30 isobs.y . . . . . . . . . . 11  |-  Y  =  ( 0g `  W
)
3129, 30syl6eqr 2491 . . . . . . . . . 10  |-  ( h  =  W  ->  ( 0g `  h )  =  Y )
3231sneqd 3887 . . . . . . . . 9  |-  ( h  =  W  ->  { ( 0g `  h ) }  =  { Y } )
3328, 32eqeq12d 2455 . . . . . . . 8  |-  ( h  =  W  ->  (
( ( ocv `  h
) `  b )  =  { ( 0g `  h ) }  <->  (  ._|_  `  b )  =  { Y } ) )
3424, 33anbi12d 710 . . . . . . 7  |-  ( h  =  W  ->  (
( A. x  e.  b  A. y  e.  b  ( x ( .i `  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  /\  ( ( ocv `  h ) `
 b )  =  { ( 0g `  h ) } )  <-> 
( A. x  e.  b  A. y  e.  b  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) ) )
358, 34rabeqbidv 2965 . . . . . 6  |-  ( h  =  W  ->  { b  e.  ~P ( Base `  h )  |  ( A. x  e.  b 
A. y  e.  b  ( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  /\  ( ( ocv `  h ) `  b
)  =  { ( 0g `  h ) } ) }  =  { b  e.  ~P V  |  ( A. x  e.  b  A. y  e.  b  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b )  =  { Y } ) } )
36 fvex 5699 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
376, 36eqeltri 2511 . . . . . . . 8  |-  V  e. 
_V
3837pwex 4473 . . . . . . 7  |-  ~P V  e.  _V
3938rabex 4441 . . . . . 6  |-  { b  e.  ~P V  | 
( A. x  e.  b  A. y  e.  b  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) }  e.  _V
4035, 1, 39fvmpt 5772 . . . . 5  |-  ( W  e.  PreHil  ->  (OBasis `  W )  =  { b  e.  ~P V  |  ( A. x  e.  b  A. y  e.  b  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b )  =  { Y } ) } )
4140eleq2d 2508 . . . 4  |-  ( W  e.  PreHil  ->  ( B  e.  (OBasis `  W )  <->  B  e.  { b  e. 
~P V  |  ( A. x  e.  b 
A. y  e.  b  ( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) } ) )
42 raleq 2915 . . . . . . . 8  |-  ( b  =  B  ->  ( A. y  e.  b 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) ) )
4342raleqbi1dv 2923 . . . . . . 7  |-  ( b  =  B  ->  ( A. x  e.  b  A. y  e.  b 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) ) )
44 fveq2 5689 . . . . . . . 8  |-  ( b  =  B  ->  (  ._|_  `  b )  =  (  ._|_  `  B ) )
4544eqeq1d 2449 . . . . . . 7  |-  ( b  =  B  ->  (
(  ._|_  `  b )  =  { Y }  <->  (  ._|_  `  B )  =  { Y } ) )
4643, 45anbi12d 710 . . . . . 6  |-  ( b  =  B  ->  (
( A. x  e.  b  A. y  e.  b  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } )  <->  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
4746elrab 3115 . . . . 5  |-  ( B  e.  { b  e. 
~P V  |  ( A. x  e.  b 
A. y  e.  b  ( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) }  <->  ( B  e.  ~P V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
4837elpw2 4454 . . . . . 6  |-  ( B  e.  ~P V  <->  B  C_  V
)
4948anbi1i 695 . . . . 5  |-  ( ( B  e.  ~P V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) )  <->  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
5047, 49bitri 249 . . . 4  |-  ( B  e.  { b  e. 
~P V  |  ( A. x  e.  b 
A. y  e.  b  ( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) }  <->  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) )
5141, 50syl6bb 261 . . 3  |-  ( W  e.  PreHil  ->  ( B  e.  (OBasis `  W )  <->  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) ) ) )
524, 51biadan2 642 . 2  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) ) )
53 3anass 969 . 2  |-  ( ( W  e.  PreHil  /\  B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B )  =  { Y } ) )  <->  ( W  e. 
PreHil  /\  ( B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) ) )
5452, 53bitr4i 252 1  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  B  C_  V  /\  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  B
)  =  { Y } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   {crab 2717   _Vcvv 2970    C_ wss 3326   ifcif 3789   ~Pcpw 3858   {csn 3875   dom cdm 4838   ` cfv 5416  (class class class)co 6089   Basecbs 14172  Scalarcsca 14239   .icip 14241   0gc0g 14376   1rcur 16601   PreHilcphl 18051   ocvcocv 18083  OBasiscobs 18125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-obs 18128
This theorem is referenced by:  obsip  18144  obsrcl  18146  obsss  18147  obsocv  18149
  Copyright terms: Public domain W3C validator